Question

Use the image to answer the question.

An illustration shows an unfolded box net. It is comprised of a rectangle at the center, two squares extending into three trapezoids on the top and bottom, and a trapezoid on the left of the rectangle. The top side of the rectangle is labeled in three distinct sections: starting at the left, a solid line of 3 is drawn, followed by a dashed line of 3, and then a solid line of 6. The bottom of the rectangle is labeled in three distinct sections: a solid line of 6, a dashed line of 3, and a solid line of 3. The left width of the rectangle is drawn as a dashed line, labeled as 3, which becomes the base of a shaded trapezoid. The dashed lines on the top and bottom of the rectangle are extended into squares of side 3, sides drawn in dashed lines. The three outer sides of each square become the longer bases of shaded trapezoids. The top shorter side of the trapezoid is labeled as 1.2 and the perpendicular height is 1.

Decompose the box net into simple polygons to find how much cardboard in square inches is needed to make the cube. Assume that all angles that look like right angles are right angles. All polygons that look congruent are congruent. What is the area of the shape in square inches?

Explain why the Correct Answer
is the only Correct Answer

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the total area of the box net, we need to calculate the area of each individual polygon and then add them up.

1. Rectangle: The area of the rectangle is calculated by multiplying its length and width. The length is the sum of the three sections on the top side (3+3+6 = 12) and the width is given as 3. Therefore, the area is 12 x 3 = 36 square inches.

2. Squares: There are two squares in the box net, each with a side length of 3. The area of one square is 3 x 3 = 9 square inches. Since there are two squares, the total area of the squares is 2 x 9 = 18 square inches.

3. Trapezoids: There are three trapezoids on the top and three trapezoids on the bottom. Each trapezoid has a longer base of 6, a shorter base of 3, and a height of 1. The area of a trapezoid can be calculated using the formula: 1/2 x (sum of bases) x height. Plugging in the values, we get 1/2 x (6+3) x 1 = 4.5 square inches for each trapezoid. Since there are a total of 6 trapezoids, the total area is 6 x 4.5 = 27 square inches.

Now, adding up the areas of the rectangle, squares, and trapezoids: 36 (rectangle) + 18 (squares) + 27 (trapezoids) = 81 square inches.

Therefore, the Correct Answer is 81 square inches.

This is the only Correct Answer because it is the sum of the areas of all the individual polygons that make up the box net. Each polygon has been calculated correctly based on the given dimensions, and adding them up gives us the total area needed to make the cube out of cardboard. No other combination of areas of polygons would result in the total area being different from 81 square inches.